Huh discovered that this kind of mathematics could give him what poetry could not: the ability to search for beauty outside himself, to try to grasp something external, objective and true, in a way that opened him up more than writing ever had. “You don’t think about your small self,” he said. “There’s no place for ego.” He found that unlike when he was a poet, he was never motivated by the desire for recognition. He just wanted to do math.
Hironaka, perhaps recognizing this, took him under his wing. After Huh graduated and started a master’s program at Seoul National University — where he also met Nayoung Kim, now his wife — he spent a lot of time with Hironaka. During breaks, he followed the professor back to Japan, staying with him in Tokyo and Kyoto, carrying his bags, sharing meals, and of course continuing to discuss math.
An Unexpected Discovery
Huh applied to about a dozen doctoral programs in the U.S. But because of his undistinguished undergraduate experience, he was rejected by all of them save one. In 2009, he began his studies at the University of Illinois, Urbana-Champaign, before transferring to the University of Michigan in 2011 to complete his doctorate.
Despite the challenges — living in a new country, spending time apart from Kim (she stayed at Seoul National University for her doctorate in mathematics) — Huh cherished his experiences in graduate school. He was able to dedicate himself wholly to math, and he relished the freedom of exploration that had drawn him to the subject in the first place.
He immediately stood out. As a beginning graduate student in Illinois, he proved a conjecture in graph theory that had been open for 40 years. In its simplest form, the problem, known as Read’s conjecture, concerned polynomials — equations like n4 + 5n3 + 6n2 + 3n + 1 — attached to graphs, which are collections of vertices (points) connected by edges (lines). In particular, let’s say you want to color the vertices of a graph so that no two adjacent vertices have the same color. Given a certain number of colors at your disposal, there are many ways to color the graph. It turns out that the total number of possibilities can be calculated using an equation called the chromatic polynomial (which is written in terms of the number of colors being used).
Mathematicians observed that the coefficients of chromatic polynomials, no matter the graph, always seem to obey certain patterns. First, they are unimodal, meaning they increase and then decrease. Take the previous example of a polynomial. The absolute values of its coefficients — 1, 5, 6, 3, 1 — form a unimodal sequence. Moreover, that sequence is also “log concave.” For any three consecutive numbers in the sequence, the square of the middle number is at least as large as the product of the terms on either side of it. (In the above polynomial, for instance, 62 ≥ 5 × 3.)
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Still, mathematicians struggled to prove these properties. And then, seemingly out of nowhere, along came Huh.
As a master’s student, he had studied algebraic geometry and singularity theory with Hironaka. The main objects of study in that field are called algebraic varieties, which can be thought of as shapes defined by certain equations. Intriguingly, associated to certain kinds of algebraic varieties are numbers that are known to be log concave — something Huh only knew because of the serendipitous direction his studies had taken him in. Huh’s key idea was to find a way to construct an algebraic variety such that those associated numbers were precisely the coefficients of the chromatic polynomial of the graph from the original question.
His solution stunned the math community. It was at that point that the University of Michigan, having rejected his initial application, recruited him to their graduate program.
Huh’s achievement was impressive not just because he had solved Read’s conjecture when it had seemed completely intractable for so long. He had shown that something much deeper — and geometric — was lurking beneath combinatorial properties of graphs.
Mathematicians were also impressed by his demeanor. His talks at conferences were always accessible and concrete; in speaking with him, it was clear that he was thinking both deeply and broadly about the concepts he was working with. “He was ridiculously mature for a graduate student,” said Matthew Baker, a mathematician at the Georgia Institute of Technology. After Baker met him for the first time, “I was just like, who is this guy?”
According to Mircea Mustaţă, Huh’s adviser at the University of Michigan, he required almost no supervision or guidance. Unlike most graduate students, he already had a program in mind, and ideas about how to pursue it. “He was more like a colleague,” Mustaţă said. “He already had his own way of looking at things.”
Many of his collaborators note that he’s incredibly humble and down-to-earth. When he learned he’d won the Fields Medal, “it didn’t really feel that good,” Huh said. “Of course you are happy, but deep down, you’re a little bit worried that they might eventually figure out that you’re not actually that good. I am a reasonably good mathematician, but am I Fields Medal-worthy?”
Escape From Space
Graphs are actually just one type of object that can define more general structures called matroids. Consider, for example, points on a two-dimensional plane. If more than two points lie on a line in this plane, you can say that those points are “dependent.” Matroids are abstract objects that capture notions like dependence and independence in all sorts of different contexts — from graphs to vector spaces to algebraic fields.
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Just as graphs have chromatic polynomials associated with them, there are equations called characteristic polynomials attached to matroids. It was conjectured that the polynomials for these more general objects should also have coefficients that are log concave. But the techniques Huh used to prove Read’s conjecture only worked for showing log concavity for a very narrow class of matroids, such as the matroids that arise from graphs.
With the mathematician Eric Katz, Huh broadened the class of matroids such a proof could apply to. They followed a recipe of sorts. As before, the strategy was to start with the object of interest — here, a matroid — and use it to construct an algebraic variety. From there, they could extract an object called a cohomology ring and use some of its properties to prove log concavity.
There was just one problem. Most matroids don’t have any sort of geometric foundation, which means there’s not actually an algebraic variety to associate to them. Instead, Huh, Katz and the mathematician Karim Adiprasito figured out a way to write down the right cohomology ring straight from the matroid, essentially from scratch. They then showed, using a new set of techniques, that it behaved as if it had come from an actual algebraic variety, even though it hadn’t. In doing so, they proved log concavity for all matroids, resolving the problem known as Rota’s conjecture once and for all. “It’s pretty remarkable that it works,” Baker said.
The work showed that “you don’t need space to do geometry,” Huh said. “That made me really fundamentally rethink what geometry is.” It would also guide him toward a host of other problems, where he continued to push that idea further, allowing him to develop an even broader range of methods.
But for all the specificity the work requires, building the right cohomology ring requires massive amounts of guesswork and groping around in the dark. It was an aspect of the work that Huh particularly enjoyed. “There is no guiding principle … no clearly defined goal,” he said. “You just have to make a guess.”
